The spectrum of pure dS$_3$ gravity in the static patch

TL;DR

This paper explores the quantum spectrum of three-dimensional de Sitter gravity in the static patch, revealing a discrete, bounded spectrum with a simple sum over saddle points that reproduces expected thermodynamic properties.

Contribution

It introduces a novel Lorentzian path integral approach with complex saddle points, connecting the static patch spectrum to the wave function at future infinity in de Sitter space.

Findings

The entropy matches the Bekenstein-Hawking formula.

The spectrum in the spin-zero sector is discrete, bounded, and includes a dense set of states.

Negative contributions to the trace are arranged to be invisible in smooth observables.

Abstract

We consider the quantum mechanical description of the de Sitter static patch in three-dimensional general relativity. We consider a Lorentzian path integral that conjecturally computes the Fourier transform of the spectrum of the static patch Hamiltonian. We regulate a saddle point for this integral by a complex deformation that connects it to future infinity. Our computation is thus closely connected with the wave function of de Sitter gravity on a torus at future infinity. Motivated by this, we identify an infinite number of saddle points that contribute to our Lorentzian path integral. Their sum gives a surprisingly simple result, which agrees with the expected features of the de Sitter static patch. For example, the thermal entropy, evaluated at the de Sitter temperature, agrees with the Bekenstein-Hawking formula. We also obtain a spectrum in the spin-zero sector, which is bounded, discrete, and has an integer degeneracy of states. It includes a dense spectrum of states, making both positive and negative contributions to the trace, arranged in such a way that negative contributions are invisible in the computation of any smooth observable. Nevertheless, several mysteries remain.